? tc?s x1?s￥à Q 1.X?/ f ?·? uW  V¨?l p/ s (1) Rba xdx(0 < a < b) (2) Rba kdx(k ^è ? (3) R2?1 x2dx (4) R10 axdx(a 6= 1;a > 0). 2 !f(x)[a+c;b+c] V￡ üf(x+c)[a;b]  V O Z b a f(x+c)dx = Z b+c a+c f(x)dx: 3 ! f(x) = 8 < : 1;x = c;c 2 (a;b); 0;x 2 [a;c)[(c;b]; p￡Rba f(x)dx = 0. 4 ?f ?f(x)[a;b]  V s ^I?[a;b] =μK?? ? Mf(x)￥′ P ??1 6Bf ?f?(x)￡ üf?(x)9[a;b]  Vi O s ˉ1I. x2?s￥'?é 1. !f(x)[a;b] ??f(x) ? 0f(x)??1 ,￡ ü Z b a f(x)dx > 0: 1 2. !f(x)[a;b] ??Rba f2(x)dx = 0￡ üf(x)[a;b] ?1 ,. 3.  è a üf2(x)[a;b] V?f(x)[a;b]? V. 4.1?/ ò?s￥vl (1) R10 xdx , R10 x2dx (2) R … 20 xdx, R … 20 sinxdx (3) R?1?2 (13)xdx, R10 3xdx . 5￡ ü/ ?? T ! ?ó￥si (1) 1 ?R10 ex2dx ? e (2) 1 ?R … 20 sinx x dx ? … 2 (3)…2 6R … 20 dxp 1?12 sin2 x 6 …p 2 (4) 3pe ?R4e0 lnxdxpx ? 6. 6￡ ü (1) limn!1R10 xn1+xdx = 0 (2) limn!1R … 20 sinn xdx = 0. 7 !f(x);g(x)[a;b] ??￡ ü lim ?!0 nX i=1 f(?i)g( i)￠xi = Z b a f(x)g(x)dx ?a = x0 < x1 < ￠￠￠ < xn = b;￠xi = xi ? xi?1;?i; i 2 [xi?1;xi](i = 1;2;￠￠￠ ;n);? = max1?i?nf￠xig. 2 8. !f0(x)[a;b] ?? Of(a) = 0 p￡ flfl flfl fl Z b a f(x)dx flfl flfl fl? (b?a)2 2 maxa?x?b flflf0(x)flfl: 9. !0 < – < 1 p￡ limn!1 R1 – (1?t 2)ndt R1 0 (1?t2)ndt = 0: 10(1) !f(x)[a;b]  ?? O[a;b]  ?B ??f ?g(x) ( μRba f(x)g(x)dx = 0￡ üf(x) · 0;x 2 [a;b]. (2) !f(x)[a;b]  ?? O ?μ *t[a;b]  ?@?FHqg(a) = g(b) = 0￥ ??f ?g(x)μRba f(x)g(x)dx = 0.￡ ü[a;b] ]" μf(x) · 0. 11 !f(x);g(x)[a;b] ?? p￡ flfl flfl fl Z b a f(x)g(x)dx flfl flfl fl? sZ b a f2(x)dx￠ sZ b a g2(x)dx 7 O?|? ?? O??g(x) = ?f(x)(f(x) = ?g(x)) ??1è ?b 12. !f(x);g(x)[a;b] ?? p￡ sZ b a [f(x)+g(x)]2dx ? sZ b a f2(x)dx+ sZ b a g2(x)dx 7 O?|? ?? O??g(x) = ?f(x)(? ? 0è ?). 13. !f(x)[0;1] ??f(x) ? fi > 0 p￡ Z 1 0 1 f(x)dx ? 1R 1 0 f(x)dx : 14. !y = ’(x)(x ? 0) ^?ì??9F￥ ??f ?’(0) = 0;x = `(y) ^ ?￥Qf ?￡ ü Z a 0 ’(x)dx+ Z b 0 `(y)dy ? ab(a ? 0;b ? 0): 3 15.¨Bá ???l￡ (1) f(x) = 3px[0;1]  ^Bá ??￥ (2) f(x) = sinx(?1;+1)  ^Bá ??￥ (3) f(x) = x2[a;b] Bá ???(?1;+1) ?Bá ?? (4) f(x) = sinx2(?1;+1) ?Bá ??. x3±s'? ? 1.9 ?/ ?s (1) R…0 cos2 xdx (2) Ra0 pa?xdx (3) R…0 p 1?sin2 xdx (4) R?3?4 dxxpx2?4 (5) R21 lnxx dx (6) Re1 e jlnxjdx 2 p/ K (1) limn!1 nP k=1 1 n sin k… n (2) limn!1?1n + 1n+1 +￠￠￠+ 12n￠ (3) limn!1 nP k=1 k n2 (4) limn!1 1n npn(n+1)￠￠￠(2n+1) 3 ?f(x) ?? pF0(x) 4 (1) F(x) = Rx20 f(t)dt (2) F(x) = Rbx f(t)dt (3) F(x) = Rx3x et2dt 4 p/ K (1) limn!1 Rx 0 cost 2dt x (2) limn!1 ?Rx 0 e t2dt ·2 Rx 0 e 2t2dt 5 !f(x)[0;+1) ?? O???9 p￡f ? F(x) = 1x Z x 0 f(t)dt (0;+1)  ?? O???9b x4?s￥9 ? 1.9 ?/ ?s (1) R21 (x+1)(x2?3)3x2 dx (2) R10 x2+1x4+1dx (3) R 1 5 ?15 x p2?5xdx (4) R94 (px+ 1px)dx (5) R10 p4?x2dx (6) Ra0 x2pa2 ?x2dx (7) R … 20 sinmxcosnxdx (8) R10 dx(x2?x+1)3=2 5 (9) R30 xdx1+p1+x (10) R40 x(x+px)dx (11) R … 21 cosx 1+sin2 xdx (12) R10 e px dx (13) R10 xarctanxdx (14) R2…0 x2 cos2 xdx (15) R…?… x2 cos2 xdx (16) R pln2 0 x 3e?x2dx (17) R…?… sinmxcosnxdx (18) Ra0 x2 q a?x a+xdx(a > 0) (19) R2a0 px2?a2 x4 dx (20) R 1p 5 0 x3(1?5x2)10dx 29 ?/ ?s (1) R … 20 sin9 xdx (2) R…0 sin5 xdx (3) R2…0 cos6 xdx (4) R 3… 20 cos7 xdx (5) Ra0 (a2 ?x2)ndx (6) R10 (1?x2)6dx 6 3￡ ü ??￥ f ?￥B Mef ?￥1 }f ? ??￥ }f ?￥e f ??μ OoμB?1 f ?. 4 !f(x) ? U uW  ^ ??f ?￡ ü (1) R … 20 f(sinx)dx = R … 20 f(cosx)dx (2) R…0 xf(sinx)dx = …2 R…0 f(sinx)dx (3) Ra1 f(x2 + a2x2)dxx = Ra21 f(x+ a2x )dx2x (4) Ra0 x3f(x2)dx = 12 Ra20 xf(x)dx(a > 0) 59 ?sR …20 sinxcosx+sinxdx. 6 ?¨s?sE￡ ü Z b a f(u)(x?u)du = Z x 0 ( Z u 0 f(x)dx)du: 7. !f00(x)[a;b] ?? Of(a) = f(b) = 0 p￡ (1) Rba f(x)dx = 12 Rba f00(x)(x?a)(x?b)dx (2) flfl flRba f(x)dx flfl fl? (b?a)312 maxa?x?bjf00(x)j 8 !f(x)x > 0 H ?? ?ia;b > 0s′ Z ab a f(x)dx Daí1 p￡f(x) = cx c1è ?. 9 !f(x) ?BμK uW  Vs O limx!1f(x) = l p￡ limx!1 1x Z x 0 f(t)dt = l 7 x5?st ??￥?¨? 1.μB ?x2a2 + y2b2 ? 1(a > b)éà #°Z_B?? ? ￡? p ￡ e￥a ?. 2.?yv B B H15/??b !B?????￥°?120m ￡ '27m??ú ￡ ?3m1ü ￡D9 ? X?× ? ?T￥?b 3.  ￡ o￥C ê ^B0? ?6m/?2mú10m p ￡9 ? H C ê ?1￥ ?b ! ￡￥1×11000kg=m3. 4.??1r￥ o? ? ￡? ?D ￡ ?M¤ o￥1×11C| oV ￡? |1T ?$ 5.ü?T ?é ?3￥ ?D?T￥ %é??1bX?1kg￥ ? ? P?T %é1cmùü?T ?é10cm1T ?$ 6.μBé1a￥%? ?ò?)￥L áDM  B ?￥  ? ü Z??1 pN%?￥ ü ( á. x6?s￥í ?9 ? 1.X?R10 dx1+x2 = …4 küs uW[0;1]s?10?ssY¨0? T ? ?tL T9 ?…￥í ?′ú ??l ??a ?ê. 2.üs uW10?s¨ ?tL T9 ?/ s￥í ?′ú ? ?l ??a ?ê (1) R10 p1?x3dx (2) R21 dxx . 8